Spectral analysis for transmission eigenvalue problems with and without the complementing conditions

The interior transmission eigenvalue problem is a system of partial differential equations equipped with Cauchy data on the boundary: the transmission conditions. This problem appears in the inverse scattering theory for inhomogeneous media when, for some frequency, the incident wave is not scattered by the medium. The relative Cauchy problem associated to the interior transmission eigenvalue problem plays also a crucial role in the stability of mathematical models describing phenomena using negative-index metamaterials. The first chapter is devoted to the analysis of the interior transmission eigenvalue problem in the acoustic setting. We establish the Weyl law for the interior transmission eigenvalues and the completeness of the generalized eigenfunctions for a system without complementing conditions, i.e., the two equations of the system have the same coefficients for the second order terms, and thus being degenerate. These coefficients are allowed to be anisotropic and are assumed to be two times continuously differentiable. One of the keys of the analysis is to establish the well-posedness and the regularity in $L^p$-scale for the relative Cauchy problem. To this end we mainly use the Fourier analysis, which allows obtaining results for non-smooth coefficients. As a result, we extend and rediscover known results for which the coefficients for the second order terms are required to be isotropic and smooth, but using a new approach. The second chapter is related to the interior transmission eigenvalue problem in the electromagnetic setting. Cakoni and Nguyen recently proposed very general conditions on the coefficients of Maxwell equations for which they established the discreteness of the set of interior transmission eigenvalues and studied their location. In this chapter, we establish the completeness of the generalized eigenfunctions and derive an optimal upper bound for the counting function under these conditions and assuming additionally that the coefficients are twice continuously differentiable. The approach is based on the spectral theory of Hilbert-Schmidt operators. Eventually, we present in chapter 3 partial results on the interior transmission eigenvalue problem in the electromagnetic setting under non-complementing conditions. We only prove a key point of our analysis leading to a priori estimates of the solutions to the relative Cauchy problem.